Bounds on the localization number

TL;DR

This paper investigates the localization number in graphs, establishing bounds related to chromatic number and degeneracy, and determining specific values for classes like outerplanar graphs and hypercubes.

Contribution

It proves an upper bound on the localization number based on chromatic number, confirms the tightness of degeneracy bounds, and computes the localization number for certain graph classes.

Findings

Graphs with localization number ≤ k have degeneracy < 3^k.

The localization number is at most 2 for outerplanar graphs.

The localization number of hypercubes is determined up to an additive constant.

Abstract

We consider the localization game played on graphs, wherein a set of cops attempt to determine the exact location of an invisible robber by exploiting distance probes. The corresponding optimization parameter for a graph $G$ is called the localization number and is written $\zeta (G)$. We settle a conjecture of \cite{nisse1} by providing an upper bound on the localization number as a function of the chromatic number. In particular, we show that every graph with $\zeta (G) \le k$ has degeneracy less than $3^k$ and, consequently, satisfies $\chi(G) \le 3^{\zeta (G)}$. We show further that this degeneracy bound is tight. We also prove that the localization number is at most 2 in outerplanar graphs, and we determine, up to an additive constant, the localization number of hypercubes.